Parallel Objects and Field Equations

TL;DR

This paper generalizes the concept of parallel geometric objects to incorporate change and conservation, proposing a framework that encompasses key physical field equations like Maxwell and Yang-Mills.

Contribution

It introduces a new mathematical framework for representing physical objects and their dynamics, unifying change and conservation in field equations.

Findings

Most important physical equations follow the proposed rule.

Extended Maxwell and Yang-Mills equations are analyzed within this framework.

The approach suggests a new way to search for field equations in physics.

Abstract

This paper considers a generalization of the existing concept of parallel (with respect to a given connection) geometric objects and its possible usage as a suggesting rule in searching for adequate field equations in theoretical physics. The generalization tries to represent mathematically the two-sided nature of the physical objects, the {\it change} and the {\it conservation}. The physical objects are presented mathematically by sections $\Psi$ of vector bundles, the admissible changes $D\Psi$ are described as a rsult of the action of appropriate differential operators $D$ on these sections, and the conservation propertieis are accounted for by the requirement that suitable projections of $D\Psi$ on $\Psi$ and on other appropriate sections must be zero. It is shown that the most important equations of theoretical physics obey this rule. Extended forms of Maxwell and Yang-Mills equations are also considered.